Strictly speaking, several differences between
the closely related CHSP and PTQM unitary models
are non-trivial [1]}. This fact happened to be obscured
by the slightly misleading current terminology.
Pars pro toto, let us mention that the
Hamiltonian operators \(H\)  which are
all required to possess the real spectrum and which are all
required self-adjoint in the correct physical Hilbert space
\({\cal H}_{phys}\) 
are still called non-Hermitian in the literature.
Presumably, the reasons are psychological:
All of the
necessary two-space calculations are, naturally,
performed in just one of the frames, viz., in
the auxiliary and manifestly unphysical Hilbert space
\({\cal H}_{math}\) 
in which the Hamiltonians
really are  non-Hermitian.
